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Alexander Gruber♦Oct 7 '12 at 4:25

$[G^{(n-1)},G^{(n-1)}]=G^{(n-2)}$, so $[G^{(n-1)},G^{(n-1)}]$ goes to the identity in $G/G^{(n-2)}$. Now can you prove that $G^{(k)}/G^{(n-1)}=(G/G^{(n-1)})^{(k)}$?
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Alexander Gruber♦Oct 7 '12 at 5:35